On an analogy between two plane problems of mechanics in a ring

A. O. Kazakova

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 2, pp. 211-227

Résumé

The article discusses an analogy between two plane problems of continuum mechanics: the hydrodynamic problem of the motion of a viscous fluid enclosed between two rotating cylinders, and the plane problem of the theory of elasticity in stresses created in a tube by a constant normal external pressure. In both problems, the solution domain is a ring; within the framework of this paper, two cases are considered: a concentric and an eccentric ring. In the first part of the article, an analogy is constructed for the case of a concentric ring; it is shown that in this case the solutions to the problems in question are expressed by functions of the same type. The second part of the article presents an attempt to build a direct analogy for the case of an eccentric ring and identifies the problems that arise. The third part of the article is aimed at establishing the stress state in the eccentric ring corresponding to the biharmonic stress function constructed by analogy with the hydrodynamic problem under study, taking into account the conditions for the single-valued displacements. As a result of the study, it can be concluded that an analogy between the problems under consideration can be established, but only taking into account the mechanical features of each of them. In the case of a concentric ring, there is a direct analogy.

Détail

Mots-clés : viscous fluid, annular domain
Keywords: plane problem of the theory of elasticity, bipolar coordinates, stream function, stress function.